On Koebe-type functions for harmonic quasiconformal mappings

Abstract

This paper studies a class of Koebe-type harmonic quasiconformal functions. It is motivated by the shear construction of Clunie and Sheil-Small [Ann. Acad. Sci. Fenn. Ser. A I Math. 9: 3--25, 1984] and the harmonic quasiconformal Koebe function. Equivalent univalence conditions, pre-Schwarzian and Schwarzian norms, coefficient inequalities, as well as growth and area theorems for this family of functions are established. These findings improve several previously known results.